Cantilever with a point load at the free end\nFor a cantilever beam of length l carrying a point load W at the free end, what is the maximum deflection?

Introduction / Context:
Cantilever beams are common in brackets, balconies, and machine elements. Knowing standard deflection formulas allows quick checks on stiffness and serviceability.

Given Data / Assumptions:

• Cantilever length l, flexural rigidity E I constant.
• Point load W applied at the free end.
• Small deflection, linear elastic behavior.

Concept / Approach:
From Euler–Bernoulli beam theory, slope and deflection are obtained by integrating the curvature equation M / (E I) = d^2 y / d x^2 with appropriate boundary conditions (zero deflection and slope at the fixed end).

Step-by-Step Solution:
Bending moment diagram for a free-end point load: M(x) = −W (l − x).Integrate twice and apply y(0) = 0 and dy/dx at x = 0 equals 0.Resulting maximum deflection at the free end: y_max = W l^3 / (3 E I).

Verification / Alternative check:
Compare with UDL on cantilever (y_max = w l^4 / (8 E I)); magnitudes are consistent for typical l and loadings.

Why Other Options Are Wrong:
W l^3 / (48 E I) is associated with simply supported configurations; W l^2 / (2 E I) has incorrect dimensions; W l^4 / (8 E I) corresponds to UDL on a cantilever; 2 W l^3 / (3 E I) is off by a factor of 2.

Common Pitfalls:
Forgetting boundary conditions at the fixed end; confusing cantilever formulas with simply supported ones.

Final Answer:
W l^3 / (3 E I)